How to make the edges of an object blur (to alpha)?
To Blur the Edges you could put a masking Texture (BW) into the fac of a mix node that mixes your prefered shader with an transparent shader.

I did it with a generated radial texture, but you could do it also with a premade texture, from your image editing software.
If you want a blur-node in blender look at Bashis work: https://www.blendswap.com/user/bashi

my Node-Tree:


Edit1:
for simple shapes like rect and circle it would be possible via ramp and mapping
for complex shapes you could do it possibly via python but its not simple!

So i think the best way for you is to do it with external software like photoshop. I prepared a workflow that creates the map from your shape so you dont have to approximate via drawing:  


Render with orthogonal camera topdown, your shape with emission shader and black as color. Worldbackground transparent, so the edge filter from step 3 will work.  
unwrap your Mesh-shape from within the orthogonal camera with the option "Project from View" (not with bounds).  
Put the rendered image from step one into your image processing software like photoshop and apply an edge-filter like "inner glow" with white as glow color and play with the settings as you wish and add white Background  
Import image from step 3 to Blender as fac in Mixshader from my original post and use "UV" in TextureCoordinate node. Because of step 2 your image aligns perfectly.